What is the minimum number of degrees that a square can be rotated before it carries onto itself

A regular octagon has eight sides of equal length and eight equal angles. Thus, it must be unchanged by a rotation of 360°/8 = 45 °, just as a regular triangle, for instance, is unchanged by a 120° degree rotation. Any integer multiple of this must also carry the octagon onto itself. Thus, a 90° rotation, which is twice this, will also work.

Rotational symmetry of a regular polygon is based on the number of vertices it has. If the figure is rotated a full 360 degrees, each vertex lands exactly on top of where it started. Since you have 5 vertices, your starting vertex can land in 5 different starting places and still come up with the same figure. So if you take 360/5 you get 72 degree rotations to each consecutive vertex.

This is saying that a rotation of 72, 144, 216,288 and 360 will result in the same figure. This is hard to explain without a visual, but I hope you get what I'm trying to say.

Assuming "remains unchanged" means the rotated pentagon will look like the original figure position... We can divide the pentagon (5 sides) into five inscribed triangleswith central vertex angles of 72 degrees. 360degrees ÷ 5 = 72degrees....complete rotation is 360 degrees So each rotation of the pentagon of 72 degrees will be identical

to the figure in its initial position before rotation.

Assuming "remains unchanged" means the rotated pentagon 

will look like the original figure position...

We can divide the pentagon (5 sides) into five inscribed triangles

with central vertex angles of 72 degrees.  

360degrees ÷ 5 = 72 degrees....complete rotation is 360 degrees

So each rotation of the pentagon of 72 degrees will be identical

to the figure in its initial position before rotation.